Problem: Simplify the following expression: $y = \dfrac{-4x^2+5x+21}{x - 3}$
Explanation: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-4)}{(21)} &=& -84 \\ {a} + {b} &=& &=& {5} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-84$ and add them together. Remember, since $-84$ is negative, one of the factors must be negative. The factors that add up to ${5}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-7}$ and ${b}$ is ${12}$ $ \begin{eqnarray} {ab} &=& ({-7})({12}) &=& -84 \\ {a} + {b} &=& {-7} + {12} &=& 5 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({-4}x^2 {-7}x) + ({12}x +{21}) $ Factor out the common factors: $ x(-4x - 7) - 3(-4x - 7)$ Now factor out $(-4x - 7)$ $ (-4x - 7)(x - 3)$ The original expression can therefore be written: $ \dfrac{(-4x - 7)(x - 3)}{x - 3}$ We are dividing by $x - 3$ , so $x - 3 \neq 0$ Therefore, $x \neq 3$ This leaves us with $-4x - 7; x \neq 3$.